I'd need to see a schematic, but any trace that is going to carry that high current needs to be "beefed up" or widened so that it won't burn up on you. If you are sure the maximum current will be 1A, then 2A is fine, but the question you should ask is: "how confident are you that the maximum current is 1A?" The general rule in engineering is if you have very little confidence, use a factor of 4, and if you are very confident use a factor of like 1.5 to 2.
Personally, I would multiple by two, and design it for 2A, but if this was not being mass produced I would leave the soldermask off the trace in question and put a layer of solder on it when I was building the circuit. This layer of solder will increase the maximum current and reduce the resistance of the trace. Essentially you would be increasing the "copper thickness" of 1 oz/ft^2 for that trace only.
Again, only the traces that will carry high current need to be modified.
When you ask about the ground plane being as thick as possible, do you mean the area of it as big as possible? If so, then yes the ground plane should be as large as you can make it without sacrificing performance or price. I put ground everywhere there isn't a trace on my boards (fill).
And always remember, voltage appears across and current flows through.